A Partial Solution to an Open Problem on Super Edge‐Magic Total Labelings of (4, 3)‐Cycle Books Graphs

Let H be a graph with h vertices and e edges. An edge‐magic‐total labeling of graph H is a bijective function g : V ( H ) ∪ E ( H )⟶{1, 2, …, h + e } such that g ( a ) + g ( a , b ) + g ( b ) = k , ∀( a , b ) ∈ E ( H ). When g : V ( H )⟶{1, 2, …, h }, it is termed a super edge‐magic total labeling. We introduce the concept of a ( c , d )‐cycle book graph, denoted by Book[( c , α ), ( d , β ), 2], constructed from α copies of cycles of order c and β copies of cycles of order d , each sharing a common path P 2 . The super edge‐magic total labelings problem of Book[( c , α ), ( d , β ), 2] remains unsolved. In this study, we contribute a partial solution to the super edge‐magic total labelings of Book[(4, α ), (3, β ), 2]. Specifically, we establish that the graph Book[(4, α ), (3, β ), 2] admits a super edge‐magic total labeling for all α , where α is an even positive integer and β = 1, 2. Using the super edge‐magic total labelings of Book[(4, 26), (3, 2), 2], we give an illustration how to encrypt and decrypt a secret message in cryptography by the Rivest, Shamir, and Adleman (RSA) algorithm.